On the Use of Delay Equations in Engineering Applications

Kyrychko, Yuliya N.; Hogan, S. J.

doi:10.1177/1077546309341100

Cited by 96 publications

(64 citation statements)

References 72 publications

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“…Section 5 is devoted to some illustrative examples of parabolic-type Volterra partial integro-differential equations on this topic. The paper concludes with a discussion of future developments [5], [6].…”

Section: Introductionmentioning

confidence: 99%

A Numerical Solution of Parabolic-Type Volterra Partial Integro-differential Equations by Laguerre Collocation Method

Gürbüz¹,

Sezer²

2017

IJAPM

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Partial integro-differential equations occur in many fields of science and engineering. Besides, the class of parabolic-type differential equations is modelled in compression of poro-viscoelastic media, reaction-diffusion problems and nuclear reactor dynamics. In recent years, most mathematical models used in many problems of physics, biology, chemistry and engineering are based on integral and integro-differential equations. In this work, we propose a new effective numerical scheme based on the Laguerre matrix-collocation method to obtain the approximate solution of one dimensional parabolic-type Volterra partial integro-differential equations with the initial and boundary conditions. The presented method reduces the solution of the mentioned partial integro-differential equation to the solution of a matrix equation corresponding to system of algebraic equations with unknown Laguerre coefficients. Also, some numerical examples together with error estimation are presented to illustrate the validity and applicability of the proposed scheme.

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Section: Introductionmentioning

confidence: 99%

A Numerical Solution of Parabolic-Type Volterra Partial Integro-differential Equations by Laguerre Collocation Method

Gürbüz¹,

Sezer²

2017

IJAPM

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“…In fact, the delay should be taken into consideration at the design stage to avoid the risk of instability, see Refs. [10,11].…”

Section: Introductionmentioning

confidence: 99%

On Nonoscillatory Solutions of Two-Dimensional Nonlinear Dynamical Systems

Akın¹,

Öztürk²

2017

Dynamical Systems - Analytical and Computational Techniques

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During the past years, there has been an increasing interest in studying oscillation and nonoscillation criteria for dynamical systems on time scales that harmonize the oscillation and nonoscillation theory for the continuous and discrete cases in order to combine them in one comprehensive theory and eliminate obscurity from both. We not only classify nonoscillatory solutions of two-dimensional systems of first-order dynamic equations on time scales but also guarantee the existence of such solutions using the Knaster, Schauder-Tychonoff and Schauder's fixed point theorems. The approach is based on the sign of components of nonoscillatory solutions. A short introduction to the time scale calculus is given as well. Examples are significant in order to see if nonoscillatory solutions exist or not. Therefore, we give several examples in order to highlight our main results for the set of real numbers R, the set of integers Z and q N 0 = {1, q, q 2 , q 3 , …}, q >1, which are the most well-known time scales.

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“…Delays arise whenever signals are transmitted over a finite distance from one point to another, so accurate models of communication systems inevitably should include delay differential equations. In their paper Kyrychko and Hogan [9] provide a vast array of applications of delay differential equations in engineering. Some applications worth highlighting include laser dynamics [11] and the related secure communication techniques using chaos [10].…”

Section: Introductionmentioning

confidence: 99%

Efficient computation of delay differential equations with highly oscillatory terms

et al. 2012

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Abstract. This paper is concerned with the asymptotic expansion and numerical solution of systems of linear delay differential equations with highly oscillatory forcing terms. The computation of such problems using standard numerical methods is exceedingly slow and inefficient, indeed standard software is practically useless for this purpose. We propose an alternative, consisting of an asymptotic expansion of the solution, where each term can be derived either by recursion or by solving a non-oscillatory problem. This leads to methods which, counter-intuitively to those developed according to standard numerical reasoning, exhibit improved performance with growing frequency of oscillation.

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On the Use of Delay Equations in Engineering Applications

Cited by 96 publications

References 72 publications

A Numerical Solution of Parabolic-Type Volterra Partial Integro-differential Equations by Laguerre Collocation Method

A Numerical Solution of Parabolic-Type Volterra Partial Integro-differential Equations by Laguerre Collocation Method

On Nonoscillatory Solutions of Two-Dimensional Nonlinear Dynamical Systems

Efficient computation of delay differential equations with highly oscillatory terms

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